Now, instead of thinking of the race as a sum of infinite parts, think of it as a limit of finite sub-races. For instance, in the first race Achilles runs to the one-foot mark. Achilles has run 1 1 foot in all. In the next race Achilles does the first two parts—first running 1 foot, and then a half foot. In total, Achilles has run 1 + 1/2 1.5 feet in all. The third race takes him as far as 1.75 feet, all told. Each of these sub-races is finite and well-defined; we never encounter an infinity.
